The area bounded by `y=1-|x|` and X-axis is

Options

A. 1
B. 2
C. `(1)/(2)`
D. `(3)/(4)`

Answer

To find the area bounded by the curve \( y = 1 - |x| \) and the x-axis, we can follow these steps:

### Step 1: Understand the function
The function \( y = 1 - |x| \) can be split into two cases based on the definition of the absolute value:
- For \( x \geq 0 \): \( y = 1 - x \)
- For \( x < 0 \): \( y = 1 + x \)

### Step 2: Find the points of intersection with the x-axis
To find the area, we first need to determine where the curve intersects the x-axis. This occurs when \( y = 0 \).

1. For \( x \geq 0 \):
\[ 1 - x = 0 \implies x = 1 \]
So, the point of intersection is \( (1, 0) \).

2. For \( x < 0 \):
\[ 1 + x = 0 \implies x = -1 \]
So, the point of intersection is \( (-1, 0) \).

### Step 3: Sketch the graph
The graph of the function consists of two linear segments:
- From \( (-1, 0) \) to \( (0, 1) \) (where \( y = 1 + x \))
- From \( (0, 1) \) to \( (1, 0) \) (where \( y = 1 - x \))

### Step 4: Calculate the area
The area bounded by the curve and the x-axis can be calculated as the area of the triangle formed by the points \( (-1, 0) \), \( (1, 0) \), and \( (0, 1) \).

The base of the triangle is the distance between \( (-1, 0) \) and \( (1, 0) \), which is:
\[ \text{Base} = 1 - (-1) = 2 \]

The height of the triangle is the y-coordinate of the vertex at \( (0, 1) \), which is:
\[ \text{Height} = 1 \]

The area \( A \) of the triangle is given by:
\[ A = \frac{1}{2} \times \text{Base} \times \text{Height} = \frac{1}{2} \times 2 \times 1 = 1 \]

### Conclusion
The area bounded by the curve \( y = 1 - |x| \) and the x-axis is \( 1 \) square unit.

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